I have a need to generate random planar graphs none of which have an odd cycle, i.e., bipartite graphs. I know there is a substantial two-decade literature on random planar graphs, little with which I am familiar. I know there are several models: at least those that specify an edge probability, those that depend on a random graph process, and uniform random planar graphs. I am wondering if those familiar with these and other models could suggest one that might be modified to generate even-cycled planar graphs. I don't have strict distribution requirements—randomness in any one of several senses would suffice. Thanks for your advice!

I have a need to generate random planar graphs none of which have an odd cycle, i.e., bipartite graphs. I know there is a substantial two-decade literature on random planar graphs, little with which I am familiar. I know there are several models: at least those that specify an edge probability, those that depend on a random graph process, and uniform random planar graphs. I am wondering if those familiar with these and other models could suggest one that might be modified to generate even-cycled planar graphs. I don't have strict distribution requirements—randomness in any one of several senses would suffice. Thanks for your advice!

• Generate a random planar graph, and add a vertex in the middle of each edge. Then all these newly added vertices have degree $2$.
• Select out a random subgraph of the grid graph. Then no vertex has degree greater than $4$.

I have a need to generate random planar graphs none of which have an odd cycle, i.e., even-cycled graphs. I know there is a substantial two-decade literature on random planar graphs, little with which I am familiar. I know there are several models: at least those that specify an edge probability, those that depend on a random graph process, and uniform random planar graphs. I am wondering if those familiar with these and other models could suggest one that might be modified to generate even-cycled planar graphs. I don't have strict distribution requirements—randomness in any one of several senses would suffice. Thanks for your advice!

I have a need to generate random planar graphs none of which have an odd cycle, i.e., bipartite graphs. I know there is a substantial two-decade literature on random planar graphs, little with which I am familiar. I know there are several models: at least those that specify an edge probability, those that depend on a random graph process, and uniform random planar graphs. I am wondering if those familiar with these and other models could suggest one that might be modified to generate even-cycled planar graphs. I don't have strict distribution requirements—randomness in any one of several senses would suffice. Thanks for your advice!